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Properties

Label 4-42e4-1.1-c1e2-0-14
Degree $4$
Conductor $3111696$
Sign $1$
Analytic cond. $198.404$
Root an. cond. $3.75308$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s + 8·13-s + 4·16-s + 8·25-s + 4·37-s − 16·52-s + 20·61-s − 8·64-s + 32·73-s − 16·97-s − 16·100-s + 40·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s − 8·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 4-s + 2.21·13-s + 16-s + 8/5·25-s + 0.657·37-s − 2.21·52-s + 2.56·61-s − 64-s + 3.74·73-s − 1.62·97-s − 8/5·100-s + 3.83·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.657·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3111696\)    =    \(2^{4} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(198.404\)
Root analytic conductor: \(3.75308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3111696,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.437364315\)
\(L(\frac12)\) \(\approx\) \(2.437364315\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_2$ \( 1 + p T^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 - 8 T^{2} + p^{2} T^{4} \) 2.5.a_ai
11$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.11.a_w
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.13.ai_bq
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.17.a_q
19$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.19.a_abm
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 + 40 T^{2} + p^{2} T^{4} \) 2.29.a_bo
31$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.31.a_ack
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.37.ae_da
41$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \) 2.41.a_adc
43$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.43.a_adi
47$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.47.a_dq
53$C_2^2$ \( 1 - 56 T^{2} + p^{2} T^{4} \) 2.53.a_ace
59$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.59.a_eo
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.61.au_io
67$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.67.a_afe
71$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.71.a_fm
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \) 2.73.abg_pm
79$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.79.a_agc
83$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.83.a_gk
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \) 2.89.a_ge
97$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \) 2.97.q_jy
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.353836263512789625928165935555, −9.083559597761376916998406488775, −8.604615948543984806954555397492, −8.477249383934390045871409295796, −8.031822174114039517254103200728, −7.75092787742152422470158020981, −6.96859277718540326727871250078, −6.69117887297242671851382966503, −6.34394468186259511321388436319, −5.72250297994792867685532902065, −5.53493084200743304160227091114, −4.93933073014641451114900035031, −4.60221345991105671705217530867, −3.96120476397456095112552159462, −3.67887499882034198321008981467, −3.31426668407820043049485753548, −2.63994859600334458191895790052, −1.90683177118357553204744260386, −1.03503149258991882045679211844, −0.78090895439012213134471147038, 0.78090895439012213134471147038, 1.03503149258991882045679211844, 1.90683177118357553204744260386, 2.63994859600334458191895790052, 3.31426668407820043049485753548, 3.67887499882034198321008981467, 3.96120476397456095112552159462, 4.60221345991105671705217530867, 4.93933073014641451114900035031, 5.53493084200743304160227091114, 5.72250297994792867685532902065, 6.34394468186259511321388436319, 6.69117887297242671851382966503, 6.96859277718540326727871250078, 7.75092787742152422470158020981, 8.031822174114039517254103200728, 8.477249383934390045871409295796, 8.604615948543984806954555397492, 9.083559597761376916998406488775, 9.353836263512789625928165935555

Graph of the $Z$-function along the critical line